Adiabatically-manipulated systems interacting with spin baths beyond the Rotating Wave Approximation

The Stimulated Raman Adiabatic Passage on a three-state system interacting with a spin bath is considered focusing on the efficiency of the population transfer. Our analysis is based on the perturbation treatment of the interaction term evaluated beyond the Rotating Wave Approximation, thus focusing on the limit of weak system-bath coupling. The analytical expression of the correction to the efficiency and consequent numerical analysis show that in most of the cases the effects of the environment are negligible, confirming the robustness of the population transfer.


I. INTRODUCTION
A quantum system ruled by a slowly varying Hamiltonian undergoes a dynamics known as adiabatic following of the eigenstates, based on the adiabatic theorem, according to which the populations of the instantaneous eigenstates of the Hamiltonian do not change [1,2] (details are recalled in A).This physical behavior is the key ingredient of many protocols aimed at controlling a quantum system [3][4][5][6][7].Stimulated Raman Adiabatic Passage (STIRAP) [8][9][10][11][12] represents an important and well known example of adiabatic process.The STIRAP technique was introduced to realize a complete population transfer from one state to another one by exploiting a Raman scheme involving suitable pulses with time-dependent amplitudes which couple each of the two previously mentioned states with an auxiliary one.The pulses have to be set in such a way that they should ensure the validity of the adiabatic approximation.Moreover, their specific time-dependence must be such that an eigenstate of the Hamiltonian coincides with the initial state of the system at the beginning of the process and with the target state at the end of the application of the pulses.Therefore, differently from what one could think at first glance, the total population transfer from the initial state to the target state is not due to radiative two-photon processes but plainly by an adiabatic following of an Hamiltonian eigenstate.This is particularly evident when the so called counter-intuitive sequence is considered, where the pulse which couples the auxiliary state and the target one precedes the pulse which couples the initial state to the auxiliary one.Indeed, in such a situation it is not reasonable to interpret the whole process as a (possibly virtual) photon absorption concomitant to the initial-auxiliary state transition followed by a (possibly virtual) photon emission concomitant to the auxiliary-target state transition.Moreover, it is worth observing that not only the counter-intuitive sequence works well, but that it usually works better than the so called intuitive sequence where the coupling between the auxiliary state and the initial one precedes the other coupling.(A detailed analysis of the counterintuitive sequence is given in sec.II B.) The STIRAP technique is still extensively investigated [13][14][15][16][17][18] and has been exploited in different physical contexts ranging from cold gases [19][20][21] to condensed matter [22][23][24][25][26][27][28], plasmonic systems [29,30], superconducting devices [31][32][33], trapped ions [34,35] and optomechanical systems [36].Recently, in order to improve the original technique by shortening the population transfer process, modifications of the original scheme including shortcuts to adiabaticity have been proposed [37][38][39][40][41][42].However, in this case a more complicated apparatus is required, which constitutes a disadvantage with respect to the original scheme.Indeed, implementation of shortcuts requires activation of additional interactions which somehow compensate any deviation from the adiabatic following of the eigenstates of the Hamiltonian, even in the cases where the Hamiltonian does not change slowly.In this way, the original time-dependent Hamiltonian of the system, say H(t), does not rule anymore the dynamics, since also new terms, say δH(t), are to be considered, and the total Hamiltonian H(t) + δH(t) induces a unitary evolution which coincides with an adiabatic following of the eigenstates of H(t) only.The structure of δH(t) is generally complicated and the relevant pulses need to be very precise.
As a general fact, a quantum system is subjected to the effects of noise either rising from the interaction with external systems, as for example the constituents of the environment, or related to imperfections of the apparatus.Therefore, on the one hand the fidelity of the population transfer with respect to uncertainty or fluctuations of amplitudes and phases of the pulses has been analyzed [43,44].On the other hand, the interaction with the quantized electromagnetic field has been considered, for example exploiting effective non-Hermitian Hamiltonians, which is limited to the case where the states involved in the STIRAP scheme decay toward states not involved in the procedure [45].Beyond such a specific scenario, a more general approach based on the theory of open quantum systems is required [46,47].In this line, master equations have been proposed to study the STIRAP-manipulated systems [48][49][50][51].Moreover, under suitable assumptions fitting the Davies and Spohn theory for open quantum systems ruled by time-dependent Hamiltonians [52], time-dependent master equations in the Lindblad form can be obtained from a microscopic interaction model between the atomic system and the quantized field [53,54].
Since in many cases the system to be manipulated is close to other atomic systems and interacting with them, it can happen that the interaction with the electromagnetic field is not the main source of quantum noise or, at least, not the only significant.Nitrogen Vacancies in diamond [24,27,28] or rare-earth doped crystals [22,23] are two typical examples of such a scenario.Manipulation of spin defects in magnetic materials through adiabatic followingbased techniques has been recently studied in the presence of interaction with the surrounding spins [55].Moreover, very recently, STIRAP processes on a system interacting with a spin bath has been theoretically analyzed under the assumptions that allow for the Rotating Wave Approximation (RWA) in the interaction between the three-state system and the spins of the environment [56].This study has been developed through evaluation of the unitary dynamics, in spite of the fact that master equations can be derived also for spin environments [57][58][59].
In this paper, we extend the previous study in Ref. [56] still exploiting unitary evolution of the universe but overcoming the RWA in the system-environment interaction, which makes the physical model more realistic.In fact, while the RWA implies conservation of the total number of excitation (a feature which has been extensively used in the previous analysis), it somehow excludes a variety of possible transitions.On the contrary, in this work we take into account all the terms of the system-bath interaction, thus introducing processes which can be responsible for a reduction of the population transfer efficiency.The paper is structured as follows: in the next section we describe the physical system and the relevant Hamiltonian model, also providing a brief sketch of the STIRAP technique in the ideal case and beyond, also introducing the theoretical analysis based on the perturbation approach, specifically to the truncation of the Dyson series, after two changes of picture.In sec.III we give the explicit form of the correction to the efficiency of the population transfer according to our theory and then we show predictions based on numerical calculations.Finally, in sec.IV we present an extensive discussion on the results.Two appendixes complete the presentation: in the first one the adiabatic approximation is recalled while in the second appendix we provide details about the matrix elements involved in the perturbation treatment.

A. Hamiltonian Model
The physical system we are focusing on consists of a three-state system subjected to two coherent fields and interacting with the surrounding environment consisting of a spin bath.The three-state system has two ground states (|g 1 ⟩ and |g 2 ⟩) and an excited state (|e⟩), and its free dynamics is governed by the Hamiltonian H A given below.The action of the STIRAP pulses coupling each of the two ground states with the excited one (see Fig. 1) is described by H S .In addition, the free spin-bath (an ensemble of two-state systems) is described by H B while the system-bath interaction is described by H AB which associates spin flips with atomic transitions between the excited state and each of the two ground states.Therefore, the total Hamiltonian is given by (ℏ = 1): with: where ν is the energy gap between the free excited states and the two grounds, ν ′ is the frequency of the two pulses whereas Ω m 's describe the profiles of the pulses; the natural frequency of the spins of the bath is ω, while the quantity , we have that the generator of the time evolution is: with with ∆ = ν−ν ′ the detuning between the atomic frequency and the field frequency.It is worth mentioning that, in view of the further treatment, we have split the transformed pulse Hamiltonian HS in two contributions, HS = H(0) S + H(ω) S , where the first corresponds to the so called rotating terms (characterized by the absence of fast oscillations) while the second is related to the counter-rotating terms (rapidly oscillating).For our treatment of the system-bath interaction terms in HAB (t) this separation of rotating and counter-rotating terms is not necessary.

B. Ideal STIRAP
Let us first sum up the basic of the ideal STIRAP, which corresponds to the absence of interaction with the environment (referring to our model, this condition is accomplished assuming η (m) k = 0 , ∀m, k).Moreover, assuming high atomic and field frequencies, one is legitimated to neglect the counter-rotating terms in the STIRAP Hamiltonian, so that the system can be assumed to be approximately ruled only by H ′ A + H(0) S , in this new picture usually addressed as the rotating frame.The operator can be easily diagonalized at every instant of time, and its instantaneous eigenstates are: where Such eigenstates correspond to the following eigenvalues: The standard STIRAP process aimed at transferring population from the state |g 1 ⟩ to |g 2 ⟩ is realized through a so called counterintuitive sequence, where the pulse Ω 2 precedes Ω 1 .Accordingly, at the initial time one has θ = 0 and φ = 0, so that |+⟩ = |e⟩, |0⟩ = |g 1 ⟩ and |−⟩ = |g 2 ⟩, while in the final time one has θ = π/2 and φ = 0, so that |+⟩ = |e⟩, |0⟩ = − |g 2 ⟩ and |−⟩ = |g 1 ⟩.When the pulse profiles are slowly varying functions, the hypotheses of the adiabatic theorem are satisfied and the population of each eigenstate is preserved during the evolution.Consequently, in particular, the population of the state |g 1 ⟩, initially coinciding with |0⟩, is totally transferred to the state |g 2 ⟩, which equals |0⟩ in the final time.

C. Time Evolution and Efficiency
In order to better evaluate the effects of the remaining terms of the Hamiltonian, we perform a new change of picture, applying the transformation It is worth mentioning that all the geometric phases are zero, which comes from the fact that (∂ w ⟨ϕ m (w)|) |ϕ m (w)⟩ is always an imaginary number, while the coefficients of the eigenstates are all real, which implies that all such terms are zero.Applying the transformation, we get: with with |g m (t)⟩ = U † 2 (t, t 0 ) |g m ⟩ and |e(t)⟩ = U † 2 (t, t 0 ) |e⟩.In the new picture, the generator of the time evolution is the following: and the relevant approximated dynamics can be evaluated, to the second order, by truncation of the iterated formal solution: which is essentially the truncation of the Dyson series (expressed without the chronological ordering operator) to the second order.Moreover, since we have moved to this new picture by removing the adiabatic evolution operator responsible for a perfect population transfer from |g 1 ⟩ to |g 2 ⟩, remaining in the state |g 1 ⟩ is equivalent to undergoing a perfect transition from |g 1 ⟩ to |g 2 ⟩ in the Schrödinger picture.Therefore, in the new picture the efficiency of the population transfer process through STIRAP pulses is given by the survival probability of the initial state of the three-state system: where ρ B (0) is the density operator describing the initial configuration of the bath, 1 1 B is the identity operator of the bath and T (t, t 0 ) is possibly replaced by T 2 (t, t 0 ).In our case |ψ(0)⟩ = |g 1 ⟩.In order to better understand (10), consider that the complete time evolution of the initial state |g 1 ⟩ ⟨g 1 | ⊗ ρ B (0) in the Schrödinger picture is given by: ρ

III. RESULTS
We now focus on the zero-temperature bath, which means assuming that the spin bath is initially in its ground state |{↓}⟩ = ⊗ k |↓⟩ k .Therefore, the complete initial state is |g 1 ⟩ |{↓}⟩.Since we are considering the approximation T (t, t 0 ) ≈ T 2 (t, t 0 ) according to (9), the only transitions considered in our calculations are those involving zero, one or two spin flips in the bath.This implies the following form for the efficiency: where |{↓} ↑ l {↓}⟩ is the bath state with all spins in the |↓⟩ state, except for the l-th spin, which is the |↑⟩ state, while |{↓} ↑ j {↓} ↑ l {↓}⟩ has only the l-th and j-th spins in the |↑⟩ state, all the others being in the state |↓⟩.The overlaps involving only one spin flip turn out to be zero (see Appendix B for details).The overlaps involving two spin flips involve only second-order terms and, once their squared modulus is evaluated, such terms give rise to fourth-order contributions.To make the calculation consistent with the truncation of the Dyson series to the second order, only terms up to the second order are to be kept in the probability, which gives the following expression: with k sin θ(w) η (1) We now assume that the quantities η k , first defined after (1e) without any constraint, are related in such a way that the ratios η do not depend on m, so that we can introduce 1 , define the following quantities, and recast the probability in the following form: with dw ′ e −i w ′ w E+(s)ds cos φ(w) cos φ(w) + e −i w ′ w E−(s)ds sin φ(w) sin φ(w ′ ) On the basis of ( 15), the survival probability of the initial state in the interaction picture, which corresponds to the efficiency of the population transfer in the Schrödinger picture, turns out to differ from unity by a term proportional to real part of the the integral J (t) (having the dimensions of the square of time), on which we will focus in our further analysis.Every specific correction should take into account the specific value of the square of the quantity η √ Λ (having the dimension of a frequency), which somehow is a cumulative measure of the coupling strength between the three-level system and the whole environment.It is also interesting to observe that in the RWA the counterpart of J (t) would be zero.This can be straightforwardly proven by recalculating the relevant matrix elements.Moreover, it is already well visible from the expression of J (t) where all the terms contain rapidly oscillating factors coming from the fact that all the contributions come out as matrix elements of the counter-rotating terms.
It is worth mentioning that the assumption η independent from the index m is not that restrictive as one could think, since the coupling strengths are supposed to be proportional to a function of the spin distance from the central three-state system, and this proportionality function is supposed to be the same independently from the specific transitions involved, whether |e⟩ ↔ |g 1 ⟩ or |e⟩ ↔ |g 2 ⟩.Concerning the shape of the pulses, they are usually taken as gaussian, with the peaks occurring at different times.In particular, we assume two pulses centered in ±τ and having width ∼ τ / √ 2: The process is supposed to start at time t 0 = −T (with T > τ ) and to finish at t = T .In Fig. 2 and 3 is reported the numerically calculated quantity 2ℜ{J (T )} as a function of different parameters, which allows to evaluate the efficiency of the population transfer in several regimes: the smaller is the quantity 2ℜ{J (T )}, the more efficient is the population transfer, according to (15).In all the plots we have assumed that the parameters of the pulses satisfy the following conditions, which guarantee an optimal transfer in the ideal case: Ω 0 τ = 10, T /τ = 5, ντ = 10.In Fig. 2 it is reported 2ℜ{J (T )} as a function of ω and ∆ for two values of the ratio r η , in particular r η = 1 (a) and r η = −1 (b).In both plots it is well visible that the value of 2ℜ{J (T )} is always small, never exceeding the value 0.03, and that for high values of the frequency ω the corrections become smaller and smaller.For any fixed value of ∆, one can see that varying the value of ω the quantity 2ℜ{J (T )} exhibits an oscillatory behavior, which a posteriori can be related to the presence of the phase factor exp[(ω + ν ′ )(w ′ − w)] in the integrand.In fact, in spite of the presence of other functions, the mentioned phase factor is clearly the main rapidly changing factor, whereas trigonometric functions of ϕ and θ are smoothly changing and the phase factor associated to the integral of E + changes rapidly only in the region between the two peaks.Thus, both the presence of oscillations and the vanishing of J (T ) for higher values of ν are traceable back to the oscillatory character of the counter-rotating terms.By comparing figures (a) and (b), it emerges that whether the value of the ratio parameter r η is 1 or −1 the behavior is pretty similar, though significant differences are present especially in the region corresponding to small values of ∆.In Fig. 3 the quantity 2ℜ{J (T )} is shown as a function of ω and r η for two values of the detuning ∆, in particular for ∆/ν = 0.01 (a) and ∆/ν = 0.05 (b).It is well visible that the values of the quantity in a particular parameter region are much higher than in the previous plots, reaching the value 0.8.These high values can be justified by the fact that increasing the value of r η implies increasing the amplitude of the coupling term between the three-state system and the environment.In particular, a high value of r η means dealing with a high value of η (and consequently all η (2) k ), which in turn implies a higher value of J (T ).
In order to fix the ideas about the meaning of our results, let us consider the case where η √ Λτ = 1, which means η √ Λ/Ω 0 ≪ 1, thus keeping valid the perturbation treatment.In such a case, in the greener region of Fig. 3 the correction would be about 0.8, implying an very low efficiency of about 0.2.On the contrary, in the worst region of Fig. 2, the yellow parts corresponding to 0.03, the efficiency would be of about 0.97.

IV. DISCUSSION
In this paper we have considered the effects of the interaction with a spin bath on the efficiency of a population transfer realized through a STIRAP process.Our present work is an evolution of the study reported in Ref. [56], where the system-bath interaction has been considered in the RWA.Here, in order to improve the analysis we have also considered the effects of the counter-rotating terms.Nevertheless, since the model analyzed is not exactly solvable and a numerical treatment of it would be challenging, we have faced the problem through a perturbation approach.In particular, second order corrections have been calculated by evaluating the Dyson series truncated to the second order contributions.The complete deviation of the efficiency of the population transfer from unity, according to (15), turn out to be proportional to the square of the quantity η √ Λ, which somehow plays the role of a perturbation parameter.It is the case to note that η k ) 2 ] 1/2 , which correspond somehow to effective strengths of the couplings with the environment involving |g 1 ⟩ ↔ |e⟩ transitions and |g 2 ⟩ ↔ |e⟩ transitions, respectively.Beyond this fact, according to (15), the correction to the efficiency of population transfer is proportional to the real part of the integral J (T ) defined in (16).
Remarkable, in a second order perturbation treatment under the RWA the correction to the efficiency would be zero, thus predicting a perfect population transfer.On the contrary, beyond such a approximation, deviations come up.In fact, our analysis shows that in the second perturbation treatment the counter-rotating terms in the interaction are the only ones really contributing to the corrections.Indeed, in the integrand of J (T ) appear only contributions proportional to the phase factor exp[(ω + ν ′ )(w ′ − w)] traceable back the rapidly oscillating terms in the interaction (namely, the counter-rotating terms), while no term proportional to a phase factor associated to a lower frequency ω − ν ′ (the rotating terms) is present.This clearly implies that in a model involving RWA, the correction is zero, which is perfectly in agreement with the results of Ref. [56] where the efficiency always exhibits a plateau in the weak coupling limit and up to the weak-intermediate coupling regime.
The numerical evaluation of the integral J (T ) reported in Fig. 2 and 3 shows that in the range of the parameters analyzed the effects of the environment are mainly negligible, except for specific regions where the ratio r η assumes high values, since this implies that the coupling constants η (2) k are significant.Further numerical results (not reported in this manuscript because the would essentially provide uniform tablets corresponding to zero values) show in a clear way that higher and higher values of ν imply smaller and smaller corrections, due to the fact that counter-rotating terms become more and more rapidly oscillating and thus ineffective.Summing up, negligibility of the effects of the environment is obtained for high values of ω (according to the plates shown in the manuscript) or high values of ν (according to other simulations), provided r η is not too large.Obviously the values of η √ Λ are to be kept small in order to maintain the validity of our analysis based on perturbation theory.
It is worth concluding with two more comments concerning the comparison with the results in Ref. [56].First, since our second-order perturbation treatment is valid only in the weak coupling regime, we cannot obtain corrections in the strong coupling regime where a generalized quantum Zeno effect were predicted to occur.Second, the shape of the pulses here is properly gaussian, while in Ref. [56] different expressions have been considered.Nevertheless, in spite of the more complicated analytical expression, those pulses are essentially gaussian too, in the sense that they only slightly differ form the gaussian counterparts, which makes the two situations reasonably comparable.
The complete evolution is then given by: |ψ(t)⟩ ≈ nj a nj (0) e t t 0 ⟨ φnj (s)|ϕnj (s)⟩ds e −i t t 0 En(s)ds |ϕ nj (t)⟩ , (A4) where the quantity ⟨ φnj (s)|ϕ nj (s)⟩, known as the geometric phase, is an imaginary number.Therefore, if the coefficients of the expansion of |ϕ nj (s)⟩ with respect to a given basis are all real, the phase turns out to be the sum of real numbers, which is then supposed to be equal to zero.This is the case for the eigenstates of the STIRAP Hamiltonian considered above.

FIG. 1 :
FIG. 1: STIRAP scheme: two lower states (|g1⟩ and |g2⟩) are coupled to an upper one (|e⟩) through suitable pulses.The inset shows the typical shape of the pulses.
t 0 ).Now, since the target state is |g 2 ⟩, irrespectively of the state of the bath, we need to evaluateP (t) = tr[ρ AB (t) |g 2 ⟩ ⟨g 2 | ⊗ 1 1 B ],which, after performing two cyclic permutations inside the trace functional leads to: tr[T (t, t 0